The generator matrix

 1  0  0  0  1  1  1  1  1  1  1 X^2  X  0 X^2 X^2+X X^2 X^2+X  1  1  1  X  1  X  1  1  1  0  X  1  X  1 X^2+X X^2+X  1  1  X  X  1  1  1 X^2  1 X^2+X  1 X^2  0  X  1  1  X  1  X  1 X^2+X X^2  0  1 X^2  1 X^2+X X^2+X X^2+X  1 X^2  1  1  X  X  1  0  1  1  0  1  1 X^2  1
 0  1  0  0  0  0  1 X^2+X+1  1  1 X^2+X  1  1 X^2+X  1  1  X  1 X+1 X^2+X X^2+X+1  0 X+1  1  1 X^2+X X^2+X  1  X X^2+X X^2  0  1  1 X+1 X+1 X^2+X  1  1  0 X^2+X X^2+X X+1  1 X^2+X+1  1 X^2  1  X X^2  0 X^2  1 X^2+1  0 X^2  1 X^2+X  1 X^2 X^2+X  1  0 X^2+X+1  1 X^2+X X^2+1  1  1 X^2+1  1 X^2+1  1  1  0 X^2+X+1  1  0
 0  0  1  0  0  1  0  1 X^2+1 X^2  1 X^2+X+1 X^2+1  1 X^2+X  1  1 X+1 X^2  X X+1 X^2+X X^2 X^2  1 X+1 X+1  0  1 X^2  1 X+1 X^2+X X^2+1  X X^2+X+1  0 X^2+X  1 X^2  0  1 X^2 X+1 X^2+X+1 X^2+X+1  X X^2+X  1 X^2+X+1  1 X^2+X+1  X X^2+X  0  1 X+1  X X+1  0  1 X+1  1  X  X X^2+X  X X^2 X^2  0  0 X^2+X X+1  1 X^2+X  X X+1 X^2
 0  0  0  1  1 X+1 X^2+X+1 X^2+1 X^2  X X^2+X X^2+1 X^2+X X+1 X^2+X+1  1 X^2+1 X^2 X^2+X X^2+1  0  1 X^2+1  1 X^2+1 X+1 X^2+1 X^2+X  1  X  X  0 X^2 X^2  1 X+1  1  1 X^2+X+1 X^2 X^2+X+1 X^2+1  X X^2+X X^2  1  1  0 X+1 X+1 X^2+X+1  X X^2+1 X^2  1  X  0 X^2+1  1  X X^2+X+1 X+1 X+1  X  0  0 X+1 X^2+X X+1 X^2+X+1 X^2 X+1 X^2+1 X^2+X+1  X X^2+1 X^2  0
 0  0  0  0  X  X  X  X  0  0  0 X^2+X  0 X^2+X X^2+X  X X^2+X  0 X^2 X^2+X X^2 X^2+X X^2+X  X  X X^2+X X^2+X X^2 X^2+X  0  0 X^2 X^2 X^2  X X^2+X X^2 X^2 X^2 X^2+X  0 X^2  X  X  X  0 X^2 X^2+X X^2 X^2 X^2 X^2+X  0  X  0 X^2+X  X X^2  0 X^2  X X^2  X X^2+X  X X^2+X  0 X^2 X^2  0 X^2+X X^2+X  X  0 X^2 X^2  0 X^2+X
 0  0  0  0  0 X^2 X^2  0  0 X^2 X^2 X^2 X^2  0  0  0 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2 X^2  0 X^2  0  0  0  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0  0 X^2  0  0 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2 X^2  0  0  0  0 X^2  0  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0  0

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+150x^68+428x^69+742x^70+1070x^71+1360x^72+1672x^73+2082x^74+2378x^75+2448x^76+2708x^77+2809x^78+2644x^79+2617x^80+2420x^81+2000x^82+1698x^83+1330x^84+826x^85+542x^86+372x^87+198x^88+118x^89+68x^90+28x^91+24x^92+14x^93+11x^94+2x^95+6x^97+2x^98

The gray image is a linear code over GF(2) with n=312, k=15 and d=136.
This code was found by Heurico 1.16 in 51.8 seconds.